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A030060
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Third derivative of Catalan generating function/3!.
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2
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5, 56, 420, 2640, 15015, 80080, 408408, 2015520, 9699690, 45762640, 212469400, 973496160, 4411154475, 19800295200, 88158457200, 389753179200, 1712478031110, 7483097278800, 32540135136600, 140883148005600, 607558575774150, 2610765994183776, 11182476723339600
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OFFSET
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0,1
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..200
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FORMULA
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a(n) = (2*(n+3))!/(3!*n!*(n+4)!) = (n+1)*(n+2)*(n+3)*C(n+3)/6, C(n): Catalan numbers.
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi - 5)/2.
Sum_{n>=0} (-1)^n/a(n) = 9*sqrt(5)*log(phi) - 19/2, where phi is the golden ratio (A001622). (End)
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MATHEMATICA
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Array[CatalanNumber[#] Binomial[#, 3] &, 19, 3] (* Michael De Vlieger, Dec 17 2017 *)
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PROG
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(MuPAD) combinat::catalan(n) *binomial(n, 3) $ n = 3..21 // Zerinvary Lajos, Feb 15 2007
(PARI) a(n) = (2*(n+3))!/(3!*n!*(n+4)!) \\ Andrew Howroyd, Dec 17 2017
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CROSSREFS
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Cf. A000108, A001622.
Sequence in context: A041995 A288543 A062125 * A247710 A247774 A258490
Adjacent sequences: A030057 A030058 A030059 * A030061 A030062 A030063
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KEYWORD
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nonn
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AUTHOR
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Wolfdieter Lang
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STATUS
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approved
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